Anurn contains 6 red,4 blue ,2 green and 3yellow marbles
If three marbles are picked at random, what is the probability that two are blue and one is yellow?

Options

A. `(3)/(31)`
B. `(1)/(5)`
C. `(18)/(455)`
D. `(7)/(15)`

Explanation

To solve the problem of finding the probability that when three marbles are picked at random from an urn containing 6 red, 4 blue, 2 green, and 3 yellow marbles, two are blue and one is yellow, we can follow these steps:

### Step 1: Determine the total number of marbles
First, we need to calculate the total number of marbles in the urn.

- Red marbles = 6
- Blue marbles = 4
- Green marbles = 2
- Yellow marbles = 3

Total marbles = 6 + 4 + 2 + 3 = 15

### Step 2: Calculate the total number of ways to choose 3 marbles from 15
The total number of ways to choose 3 marbles from 15 can be calculated using the combination formula:

\[ \text{Total combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Where \( n \) is the total number of items (15 marbles) and \( r \) is the number of items to choose (3 marbles).

\[ \text{Total combinations} = \binom{15}{3} = \frac{15!}{3!(15-3)!} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455 \]

### Step 3: Calculate the number of favorable outcomes
Next, we need to find the number of ways to choose 2 blue marbles and 1 yellow marble.

- The number of ways to choose 2 blue marbles from 4 blue marbles is:

\[ \text{Ways to choose blue} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

- The number of ways to choose 1 yellow marble from 3 yellow marbles is:

\[ \text{Ways to choose yellow} = \binom{3}{1} = \frac{3!}{1!(3-1)!} = 3 \]

- Therefore, the total number of favorable outcomes for choosing 2 blue and 1 yellow marble is:

\[ \text{Favorable outcomes} = \text{Ways to choose blue} \times \text{Ways to choose yellow} = 6 \times 3 = 18 \]

### Step 4: Calculate the probability
Finally, we can calculate the probability of picking 2 blue marbles and 1 yellow marble:

\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total combinations}} = \frac{18}{455} \]

### Final Answer
Thus, the probability that two marbles are blue and one is yellow is:

\[ \frac{18}{455} \]

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